Chapter 5
                                      Initial-Value Problems for Ordinary
                                                                                  Differential Equations
                                                                                                      Hung-Yuan Fan (范洪源)
                                                                                                              Department of Mathematics,
                                                                                        National Taiwan Normal University, Taiwan
                                                                                                                                  Spring 2016
                                                                                                                                                                                                            .      .    .    .   .    .   .    .   .    .   .    .    .   .   .        .        .      .     .    .
                                                                                                                                                                                                       .    .      .     .   .   .    .    .   .   .    .    .   .    .   .    .       .         .     .     .
              Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan                                                                                                    Chap . 5, Numerical Analysis (I)                                                        1/67
                                                                                                                     Section 5.1
                                                                  The Elementary Theory of
                                                                                 Initial-Value Problems
                                                                                   (初值問題的基本理論)
                                                                                                                                                                                                            .      .    .    .   .    .   .    .   .    .   .    .    .   .   .        .        .      .     .    .
                                                                                                                                                                                                       .    .      .     .   .   .    .    .   .   .    .    .   .    .   .    .       .         .     .     .
              Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan                                                                                                    Chap . 5, Numerical Analysis (I)                                                        2/67
                         Objectives
                                           Develop numerical methods for approximating the solution to
                                           initial-value problem (IVP)
                                                                                                            { dy = f(t,y),                                                      a ≤ t ≤ b,
                                                                                                                         dt
                                                                                                                        y(a) = α,                                                                                                                                                (1)
                                           where y(t) is the unique solution to IVP (1) on [a,b].
                                           Error analysis for these numerical methods.
                         Note:
                                 1
                                           The first equation in (1) is an ordinary differential equation
                                           (ODE; 常微分⽅程式).
                                 2         y(a) = α is called an initial condition (IC; 初值條件).
                                                                                                                                                                                                            .      .    .    .   .    .   .    .   .    .   .    .    .   .   .        .        .      .     .    .
                                                                                                                                                                                                       .    .      .     .   .   .    .    .   .   .    .    .   .    .   .    .       .         .     .     .
              Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan                                                                                                    Chap . 5, Numerical Analysis (I)                                                        3/67
                         Def 5.1, p. 261
                         Afunction f(t,y) satisfies a Lipschitz condition in y on a set
                         D⊆R2if∃aLipschitz constant L > 0 s.t.
                                                                                              |f(t, y ) − f(t,y )| ≤ L|y − y |,
                                                                                                                   1                                    2                                  1                  2
                         whenever (t,y ) ∈ D and (t,y ) ∈ D.
                                                                               1                                                            2
                                                                                                                                                                                                            .      .    .    .   .    .   .    .   .    .   .    .    .   .   .        .        .      .     .    .
                                                                                                                                                                                                       .    .      .     .   .   .    .    .   .   .    .    .   .    .   .    .       .         .     .     .
              Hung-Yuan Fan (范洪源), Dep. of Math., NTNU, Taiwan                                                                                                    Chap . 5, Numerical Analysis (I)                                                        4/67